For what values of x does g(x) = x−sin x have a slope of 1?

An object oscillates along a vertical line, and its position in centimeters is given by y(t) = 30(sint - 1), where t ≥ 0 is measured in seconds and y is positive in the upward direction.

Find the velocity of the oscillator, v(t) =y′(t).

13-26 Implicit differentiation Carry out the following steps.

b. Find the slope of the curve at the given point.

cos y = x; (0, π/2)

The Chain Rule for second derivatives

b. Use the formula in part (a) to calculate $\frac{d^2}{d{x}^2}\left(\sin (3x^4+5x^2+2)\right)$.

{Use of Tech} Power and energy The total energy in megawatt-hr (MWh) used by a town is given by E(t) = 400t+2400/π sin πt/12, where t≥0 is measured in hours, with t=0 corresponding to noon.

b. At what time of day is the rate of energy consumption a maximum? What is the power at that time of day?

{Use of Tech} Spring oscillations A spring hangs from the ceiling at equilibrium with a mass attached to its end. Suppose you pull downward on the mass and release it 10 inches below its equilibrium position with an upward push. The distance x (in inches) of the mass from its equilibrium position after t seconds is given by the function x(t) = 10sin t - 10cos t, where x is positive when the mass is above the equilibrium position. <IMAGE>

b. Find dx/dt and interpret the meaning of this derivative.  

109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.

b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.

$\underset{x\to 0}{\lim }\frac{\sqrt{4+\sin \left(x\right)}-2}{x}$